On the design of low-thrust transfers through L1 for the ESMO mission

On the design of low-thrust transfers through L1 for the ESMO mission

Daniel Novak1, ∗, Jeannette Heiligers2, ∗∗, and Massimiliano Vasile1, ∗∗∗

1 University of Glasgow, Aerospace Department, James Watt South Building, G12 8QQ Glasgow, UK.2 Delft University of Technology, Faculty of Aerospace Engineering, Kluyverweg 1, 2629 HS Delft, the Netherlands.

The following paper summarises the mission analysis studies performed for the phase A of the solar electric propulsion optionof the European Student Moon Orbiter (ESMO) mission. ESMO is scheduled to be launched in 2011, as an auxiliary payloadon board an Ariane 5. Hence the lack of influence on the launch date. Two methods were devised to assess efficiently widelaunch windows for the Earth-Moon transfer, they are presented below.

1 Introduction

European Student Moon Orbiter (ESMO) is a mission proposed by the Student Space Exploration and Technology Initiative(SSETI), an association cooperating with the European Space Agency (ESA) Education department. Its primary objective(Outreach mission) is to place an orbiter on a stable orbit around the Moon and to send back images of the lunar surface fromaround the poles. As a secondary objective (Science mission), some lunar gravity field mapping shall be performed with asubsatellite called Lunette [1].

The Space Advanced Research Team (SpaceART) at the University of Glasgow was selected as the primary team for themission analysis and design of ESMO. The prime objective was to assess unfavorable launch dates in 2011. As the spacecraftwill be an auxiliary payload on board the Ariane 5 launcher, the mission must be designed for a flexible launch window.During the year-long feasibility study, two methods were developed to efficiently identify and assess a large number of launchopportunities, they will be explained below.

2 Methodology

2.1 Mission assumptions

The launch of ESMO was assumed to occur in 2011 on board an Ariane 5 with an injection into a Geostationary TransferOrbit (GTO). The Outreach and the Science missions will have an initial launch mass of 180 kg and 200 kg respectively andboth will be equipped with the same gridded ion thruster providing 20 mN of thrust with a specific impulse of 3250 s.

The target lunar orbits of ESMO were chosen to satisfy the scientific and operational requirements of the mission [1]. TheOutreach mission will enter into a polar target lunar orbit (TLO) of 100 x 3500 km altitude, with a periselenium above theSouth pole, whereas the Science mission will start the experiments in a 100 x 135 km altitude polar TLO, the periseleniumbeing above the North pole. These orbits were confirmed to be stable for at least six months [2].

The trajectory is divided into three parts, a spiral-up from the Earth, a capture leg and a spiral-down around the Moon untilinjection into the target orbit. 365 spiral-ups were propagated with a tangential continuous thrust profile for each potentiallaunch day in 2011. The delicate segment to design is the capture that follows the spiral-up.

2.2 A first approach using the Circular Restricted Three-Body Problem (CRTBP)

In order to assess the capture leg, the first approach used the CRTBP of the Earth-Moon-spacecraft system and assumed thatthe trajectory was biimpulsive. This modelization corresponds to the extention of [3] to three dimensions. For each launchdate, a global optimization using differential evolution was performed to minimize the total impulse (∆Vtot) required for thespacecraft to enter and stay in the Hill sphere of the Moon. The objective function took the form ∆Vtot + k∆r where ∆r

is the distance between the end of the Earth spiral and the beginning the transfer leg and k a coefficient. A constraint onthe inclination at arrival was imposed for a direct injection into a polar lunar orbit. Each optimal biimpulsive transfer wasassumed to be the approximation of the corresponding optimal low-thrust transfer, and favorable launch dates were identifiedby comparing each optimal ∆Vtot.

It appeared that such an approach yielded acceptable transfers only for the best launch windows. Indeed, in most ofthe cases, the optimal biimpulsive transfer ended up lasting durations of one or more periods of the Earth-Moon system(27.3 days). Such trajectories are not good approximations anymore since the ellipticity of the Moon’s orbit and the solar

∗ E-mail: dnovak@eng.gla.ac.uk, Phone: +44 141 330 8470, Fax: +44 141 330 4343∗∗ E-mail: m.j.heiligers-lr@student.tudelft.nl, Phone: +31 613 963 420∗∗∗ E-mail: m.vasile@eng.gla.ac.uk, Phone: +44 141 330 6465, Fax: +44 141 330 4343

PAMM · Proc. Appl. Math. Mech. 7, 1030903–1030904 (2007) / DOI 10.1002/pamm.200700304

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Fig. 1 Moon spirals that can be linked through L1 tothe Earth’s sphere of influence within 10 days of coastarc for the Science mission (X-axis: day of arrival intoTLO since 1/1/2011, Y-axis: RAAN of TLO).

Fig. 2 Value of the ranking function vs. day of ar-rival since 1/1/2011 for the optimal transfers for eachlaunch date (Dots: Outreach mission, circles: Sciencemission).

perturbations are not taken into account in the CRTBP for a too long period of time. When constraints on the time of flightwere imposed during the optimization, ∆Vtot turned out to be too high to approximate a low-thrust transfer.

This observation, together with the stochasticity of the results and the drawback of not entering directly into the target lunarorbit with a tangential continuous thrust profile after capture, led to the devisal of a radically different design approach.

2.3 A second approach using the ranking of potential lunar spirals

The close inspection of the mission requirements [1] led to the conclusion that there exists two degrees of freedom for choosingthe target lunar orbit: the right ascension of the ascending node (RAAN) and the arrival time. Spiral-downs were generated bypropagating backwards every potential lunar orbit with a continuous tangential thrust until the aposelenium radius reached 60000 km and further propagated backwards for 10 days without thrust, letting the Earth’s perturbations act on the spacecraft foran escape from the Moon. Only those potential lunar orbits which cross the region of L1 were deemed acceptable for the restof the study, the others were discarded (Fig. 1). A catalogue of potential captures and spiral-downs was created in this way.

The task became therefore to identify the best lunar spiral of the catalogue for each launch date by estimating the lowest∆V necessary to link the leg between the end of the Earth spiral-up and the start of the capture (called phasing leg). Thepotential lunar spirals were sorted for each launch date by a ranking function. The latter took the form

RDlaunch(Darrival, Ωarrival) =

5 · 10−8

km2(rpM − rpE)2 +

2

deg(iM − iE) +

5 · 10−3

deg2(ΩM − ΩE)2

where D, Ω, rp and i stand respectively for the date, the RAAN, the perigee radius and the inclination, and where the indicesM and E relate respectively to the beginning of the transfer leg and the end of the Earth spiral. Hence the best lunar spiralsfor each launch date could be compared through their ranking function values and therefore favorable and unfavorable launchwindows could be identified.

3 Results and Conclusion

Fig. 2 shows that for both the Outreach and the Science missions the worst launch dates lie in winter and summer due to therelative positions of the initial GTO and the Moon’s orbit. The date with the highest value of the ranking function in 2011 wasselected and the corresponding phasing leg was calculated with a low-thrust optimiser. The resulting propellant consumptionfor the full trajectory was assumed to be the worst case scenario. The Outreach mission was found to require 25.3 kg ofpropellant, whereas the Science mission 29.7 kg. A 5% modelling margin is recommended though, because the trajectoriesfor all the potential launch dates were not entirely calculated.

Although slightly less time-efficient than the approach employing the CRTBP for the capture leg, because of the generationof all the potential Moon spirals, the second proposed method allows the mission analyst to assess and compare large sets oftrajectories satisfying the mission requirements and therefore to identify convenient and inconvenient launch windows easily.Further studies would include the refinement of the ranking function and the approximation of the phasing leg through theinverse shaping method [4].

References

[1] R. Walker, ESMO Phase A Mission Requirements, 2006.[2] SpaceART, ESMO Phase A Mission Analysis Report, 2007.[3] G. Mengali, A. A. Quarta, Optimization of Biimpulsive Trajectories in the Earth-Moon Restricted Three-Body System, Journal of

Guidance, Control and Dynamics, Vol. 28(2), 209-216 (2005).[4] M. Vasile, P. De Pascale, S. Casotto, On the Optimality of a Shape-based Approach on Pseudo-Equinoctial Elements, Acta Astronau-

tica, Vol. 61, 286-297 (2007).

ICIAM07 Minisymposia – 03 Nonlinear Analysis and Dynamical Systems 1030904

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim